A topological system \((X,T)\)  is a compact metric space \(X\)  together with a self homeomorphism \(T:X\rightarrow X\) . We say that \((X,T)\)  is uniquely ergodic if there is only one \(T\) -invariant Borel probability measure on \(X\) . See any of [1]}, [2]}, [3]}, [4]}, [5]}, [6]} for background on topological systems.
